Yet, recall that (REF ) is both stabilizable and detectable which, by Theorem 3.5 of [1]}, means that so is (REF ). Thus, we must have that \(\Lambda _u(A_{11})=\Lambda _u(A_{33})=\lbrace \emptyset \rbrace \)  by virtue of the classical PBH tests (see section 3.2 of [1]}). Moreover, recall that the realizations from (REF ) are minimal, which implies that \(\mathcal {P}_u(\mathbf {G}_1)=\Lambda _u(A_1)\) . Recalling also that \(\Lambda _u(A_2)=\lbrace \emptyset \rbrace \)  along with the fact that \(\mathcal {P}_u(\mathbf {G}_1\mathbf {G}_2)=\Lambda _u(A_{22})\)  and replacing all these identities in (REF ), we get \(\mathcal {P}_u(\mathbf {G}_1\mathbf {G}_2)=\mathcal {P}_u(\mathbf {G}_1)\) .\(\Box \)
